Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.9% → 98.2%

Time: 6.9s

Alternatives: 9

Speedup: 1.2×

• 61.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 98.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(a, -3, 1\right)\right), {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (pow (+ (* a a) (* b b)) 2.0)
       (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))
      INFINITY)
   (fma
    4.0
    (fma a (fma a a a) (* (* b b) (fma a -3.0 1.0)))
    (+ (pow (hypot a b) 4.0) -1.0))
   (pow a 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= ((double) INFINITY)) {
		tmp = fma(4.0, fma(a, fma(a, a, a), ((b * b) * fma(a, -3.0, 1.0))), (pow(hypot(a, b), 4.0) + -1.0));
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0)))))) <= Inf)
		tmp = fma(4.0, fma(a, fma(a, a, a), Float64(Float64(b * b) * fma(a, -3.0, 1.0))), Float64((hypot(a, b) ^ 4.0) + -1.0));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(4.0 * N[(a * N[(a * a + a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a)))))) < +inf.0

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) + \left(-1\right)} \]

      2. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + {\left(a \cdot a + b \cdot b\right)}^{2}\right)} + \left(-1\right) \]

      3. associate-+l+ 99.9%

         \[\leadsto \color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) + \left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(-1\right)\right)} \]

      4. fma-def 99.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(4, \left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right), {\left(a \cdot a + b \cdot b\right)}^{2} + \left(-1\right)\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(a, -3, 1\right)\right), {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + -1\right)} \]

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a))))))

   1. Initial program 0.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ 0.0%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)} \]

      2. sqr-pow 0.0%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      3. sqr-pow 0.0%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      4. fma-def 0.0%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      5. distribute-lft-in 0.0%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right)} - 1\right) \]

      6. sqr-neg 0.0%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\color{blue}{\left(\left(-b\right) \cdot \left(-b\right)\right)} \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\right) \]

      7. distribute-lft-in 0.0%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(\left(-b\right) \cdot \left(-b\right)\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right) \]

   3. Simplified 4.7%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right)} \]

   4. Taylor expanded in a around inf 91.0%

      \[\leadsto \color{blue}{{a}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(a, -3, 1\right)\right), {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 2: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq \infty:\\ \;\;\;\;{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (pow (+ (* a a) (* b b)) 2.0)
       (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))
      INFINITY)
   (+
    (pow (fma a a (* b b)) 2.0)
    (+ (* 4.0 (fma (* a a) (+ a 1.0) (* (* b b) (+ 1.0 (* a -3.0))))) -1.0))
   (pow a 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= ((double) INFINITY)) {
		tmp = pow(fma(a, a, (b * b)), 2.0) + ((4.0 * fma((a * a), (a + 1.0), ((b * b) * (1.0 + (a * -3.0))))) + -1.0);
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0)))))) <= Inf)
		tmp = Float64((fma(a, a, Float64(b * b)) ^ 2.0) + Float64(Float64(4.0 * fma(Float64(a * a), Float64(a + 1.0), Float64(Float64(b * b) * Float64(1.0 + Float64(a * -3.0))))) + -1.0));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[Power[N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 + N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a)))))) < +inf.0

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ 99.9%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)} \]

      2. sqr-pow 99.9%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      3. sqr-pow 99.9%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      4. fma-def 99.9%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      5. distribute-lft-in 99.9%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right)} - 1\right) \]

      6. sqr-neg 99.9%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\color{blue}{\left(\left(-b\right) \cdot \left(-b\right)\right)} \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\right) \]

      7. distribute-lft-in 99.9%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(\left(-b\right) \cdot \left(-b\right)\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right)} \]

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a))))))

   1. Initial program 0.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ 0.0%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)} \]

      2. sqr-pow 0.0%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      3. sqr-pow 0.0%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      4. fma-def 0.0%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      5. distribute-lft-in 0.0%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right)} - 1\right) \]

      6. sqr-neg 0.0%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\color{blue}{\left(\left(-b\right) \cdot \left(-b\right)\right)} \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\right) \]

      7. distribute-lft-in 0.0%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(\left(-b\right) \cdot \left(-b\right)\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right) \]

   3. Simplified 4.7%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right)} \]

   4. Taylor expanded in a around inf 91.0%

      \[\leadsto \color{blue}{{a}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq \infty:\\ \;\;\;\;{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 3: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\\ \mathbf{if}\;t_0 \leq \infty:\\ \;\;\;\;t_0 + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))))
   (if (<= t_0 INFINITY) (+ t_0 -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double t_0 = pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 + -1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a)))))) < +inf.0

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a))))))

   1. Initial program 0.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ 0.0%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)} \]

      2. sqr-pow 0.0%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      3. sqr-pow 0.0%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      4. fma-def 0.0%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      5. distribute-lft-in 0.0%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right)} - 1\right) \]

      6. sqr-neg 0.0%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\color{blue}{\left(\left(-b\right) \cdot \left(-b\right)\right)} \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\right) \]

      7. distribute-lft-in 0.0%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(\left(-b\right) \cdot \left(-b\right)\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right) \]

   3. Simplified 4.7%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right)} \]

   4. Taylor expanded in a around inf 91.0%

      \[\leadsto \color{blue}{{a}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq \infty:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 4: 94.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+24}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.25e+24)
   (pow a 4.0)
   (if (<= a 1.35e+15) (+ (* b (* b (fma b b 4.0))) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.25e+24) {
		tmp = pow(a, 4.0);
	} else if (a <= 1.35e+15) {
		tmp = (b * (b * fma(b, b, 4.0))) + -1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.25e+24)
		tmp = a ^ 4.0;
	elseif (a <= 1.35e+15)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.25e+24], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 1.35e+15], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.25000000000000011e24 or 1.35e15 < a

   1. Initial program 41.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ 41.2%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)} \]

      2. sqr-pow 41.2%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      3. sqr-pow 41.2%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      4. fma-def 41.2%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      5. distribute-lft-in 41.2%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right)} - 1\right) \]

      6. sqr-neg 41.2%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\color{blue}{\left(\left(-b\right) \cdot \left(-b\right)\right)} \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\right) \]

      7. distribute-lft-in 41.2%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(\left(-b\right) \cdot \left(-b\right)\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right) \]

   3. Simplified 44.1%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right)} \]

   4. Taylor expanded in a around inf 94.5%

      \[\leadsto \color{blue}{{a}^{4}} \]

   if -1.25000000000000011e24 < a < 1.35e15

   1. Initial program 97.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around 0 82.2%

      \[\leadsto \color{blue}{\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
   3. Step-by-step derivation

      1. +-commutative 82.2%

         \[\leadsto \color{blue}{\left(\left(4 \cdot {b}^{2} + {b}^{4}\right) + -12 \cdot \left(a \cdot {b}^{2}\right)\right)} - 1 \]

      2. +-commutative 82.2%

         \[\leadsto \left(\color{blue}{\left({b}^{4} + 4 \cdot {b}^{2}\right)} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) - 1 \]

      3. associate-+l+ 82.2%

         \[\leadsto \color{blue}{\left({b}^{4} + \left(4 \cdot {b}^{2} + -12 \cdot \left(a \cdot {b}^{2}\right)\right)\right)} - 1 \]

      4. associate-*r* 82.2%

         \[\leadsto \left({b}^{4} + \left(4 \cdot {b}^{2} + \color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}}\right)\right) - 1 \]

      5. *-commutative 82.2%

         \[\leadsto \left({b}^{4} + \left(4 \cdot {b}^{2} + \color{blue}{\left(a \cdot -12\right)} \cdot {b}^{2}\right)\right) - 1 \]

      6. metadata-eval 82.2%

         \[\leadsto \left({b}^{4} + \left(4 \cdot {b}^{2} + \left(a \cdot \color{blue}{\left(-3 \cdot 4\right)}\right) \cdot {b}^{2}\right)\right) - 1 \]

      7. associate-*l* 82.2%

         \[\leadsto \left({b}^{4} + \left(4 \cdot {b}^{2} + \color{blue}{\left(\left(a \cdot -3\right) \cdot 4\right)} \cdot {b}^{2}\right)\right) - 1 \]

      8. *-commutative 82.2%

         \[\leadsto \left({b}^{4} + \left(4 \cdot {b}^{2} + \left(\color{blue}{\left(-3 \cdot a\right)} \cdot 4\right) \cdot {b}^{2}\right)\right) - 1 \]

      9. distribute-rgt-in 96.0%

         \[\leadsto \left({b}^{4} + \color{blue}{{b}^{2} \cdot \left(4 + \left(-3 \cdot a\right) \cdot 4\right)}\right) - 1 \]

      10. unpow2 96.0%

          \[\leadsto \left({b}^{4} + \color{blue}{\left(b \cdot b\right)} \cdot \left(4 + \left(-3 \cdot a\right) \cdot 4\right)\right) - 1 \]

      11. metadata-eval 96.0%

          \[\leadsto \left({b}^{4} + \left(b \cdot b\right) \cdot \left(\color{blue}{1 \cdot 4} + \left(-3 \cdot a\right) \cdot 4\right)\right) - 1 \]

      12. distribute-rgt-in 96.0%

          \[\leadsto \left({b}^{4} + \left(b \cdot b\right) \cdot \color{blue}{\left(4 \cdot \left(1 + -3 \cdot a\right)\right)}\right) - 1 \]

      13. associate-*l* 96.0%

          \[\leadsto \left({b}^{4} + \color{blue}{b \cdot \left(b \cdot \left(4 \cdot \left(1 + -3 \cdot a\right)\right)\right)}\right) - 1 \]

      14. distribute-lft-in 96.0%

          \[\leadsto \left({b}^{4} + b \cdot \left(b \cdot \color{blue}{\left(4 \cdot 1 + 4 \cdot \left(-3 \cdot a\right)\right)}\right)\right) - 1 \]

      15. metadata-eval 96.0%

          \[\leadsto \left({b}^{4} + b \cdot \left(b \cdot \left(\color{blue}{4} + 4 \cdot \left(-3 \cdot a\right)\right)\right)\right) - 1 \]

      16. associate-*r* 96.0%

          \[\leadsto \left({b}^{4} + b \cdot \left(b \cdot \left(4 + \color{blue}{\left(4 \cdot -3\right) \cdot a}\right)\right)\right) - 1 \]

      17. metadata-eval 96.0%

          \[\leadsto \left({b}^{4} + b \cdot \left(b \cdot \left(4 + \color{blue}{-12} \cdot a\right)\right)\right) - 1 \]

   4. Simplified 96.0%

      \[\leadsto \color{blue}{\left({b}^{4} + b \cdot \left(b \cdot \left(4 + -12 \cdot a\right)\right)\right)} - 1 \]
   5. Step-by-step derivation

      1. +-commutative 96.0%

         \[\leadsto \color{blue}{\left(b \cdot \left(b \cdot \left(4 + -12 \cdot a\right)\right) + {b}^{4}\right)} - 1 \]

      2. associate-*r* 96.0%

         \[\leadsto \left(\color{blue}{\left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right)} + {b}^{4}\right) - 1 \]

      3. sqr-pow 96.0%

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right) + \color{blue}{{b}^{\left(\frac{4}{2}\right)} \cdot {b}^{\left(\frac{4}{2}\right)}}\right) - 1 \]

      4. metadata-eval 96.0%

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right) + {b}^{\color{blue}{2}} \cdot {b}^{\left(\frac{4}{2}\right)}\right) - 1 \]

      5. pow2 96.0%

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right) + \color{blue}{\left(b \cdot b\right)} \cdot {b}^{\left(\frac{4}{2}\right)}\right) - 1 \]

      6. metadata-eval 96.0%

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right) + \left(b \cdot b\right) \cdot {b}^{\color{blue}{2}}\right) - 1 \]

      7. pow2 96.0%

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right) + \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) - 1 \]

      8. distribute-lft-out 97.9%

         \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(\left(4 + -12 \cdot a\right) + b \cdot b\right)} - 1 \]

      9. +-commutative 97.9%

         \[\leadsto \left(b \cdot b\right) \cdot \left(\color{blue}{\left(-12 \cdot a + 4\right)} + b \cdot b\right) - 1 \]

      10. fma-def 97.9%

          \[\leadsto \left(b \cdot b\right) \cdot \left(\color{blue}{\mathsf{fma}\left(-12, a, 4\right)} + b \cdot b\right) - 1 \]

   6. Applied egg-rr 97.9%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(-12, a, 4\right) + b \cdot b\right)} - 1 \]

   7. Taylor expanded in a around 0 97.9%

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(4 + {b}^{2}\right)} - 1 \]
   8. Step-by-step derivation

      1. unpow2 97.9%

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(4 + {b}^{2}\right) - 1 \]

      2. *-commutative 97.9%

         \[\leadsto \color{blue}{\left(4 + {b}^{2}\right) \cdot \left(b \cdot b\right)} - 1 \]

      3. unpow2 97.9%

         \[\leadsto \left(4 + \color{blue}{b \cdot b}\right) \cdot \left(b \cdot b\right) - 1 \]

      4. +-commutative 97.9%

         \[\leadsto \color{blue}{\left(b \cdot b + 4\right)} \cdot \left(b \cdot b\right) - 1 \]

      5. fma-def 97.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, b, 4\right)} \cdot \left(b \cdot b\right) - 1 \]

   9. Simplified 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, b, 4\right) \cdot \left(b \cdot b\right)} - 1 \]

   10. Taylor expanded in b around 0 98.0%

       \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   11. Step-by-step derivation

       1. unpow2 98.0%

          \[\leadsto \left(4 \cdot \color{blue}{\left(b \cdot b\right)} + {b}^{4}\right) - 1 \]

       2. metadata-eval 98.0%

          \[\leadsto \left(4 \cdot \left(b \cdot b\right) + {b}^{\color{blue}{\left(2 \cdot 2\right)}}\right) - 1 \]

       3. pow-sqr 97.9%

          \[\leadsto \left(4 \cdot \left(b \cdot b\right) + \color{blue}{{b}^{2} \cdot {b}^{2}}\right) - 1 \]

       4. unpow2 97.9%

          \[\leadsto \left(4 \cdot \left(b \cdot b\right) + \color{blue}{\left(b \cdot b\right)} \cdot {b}^{2}\right) - 1 \]

       5. unpow2 97.9%

          \[\leadsto \left(4 \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) - 1 \]

       6. distribute-rgt-out 97.9%

          \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)} - 1 \]

       7. +-commutative 97.9%

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b + 4\right)} - 1 \]

       8. fma-udef 97.9%

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\mathsf{fma}\left(b, b, 4\right)} - 1 \]

       9. associate-*r* 98.0%

          \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)} - 1 \]

   12. Simplified 98.0%

       \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+24}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 5: 82.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{+24}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{+14}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.15e+24)
   (pow a 4.0)
   (if (<= a 3.1e+14) (+ -1.0 (* b (* b 4.0))) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.15e+24) {
		tmp = pow(a, 4.0);
	} else if (a <= 3.1e+14) {
		tmp = -1.0 + (b * (b * 4.0));
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.15d+24)) then
        tmp = a ** 4.0d0
    else if (a <= 3.1d+14) then
        tmp = (-1.0d0) + (b * (b * 4.0d0))
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.15e+24) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 3.1e+14) {
		tmp = -1.0 + (b * (b * 4.0));
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.15e+24:
		tmp = math.pow(a, 4.0)
	elif a <= 3.1e+14:
		tmp = -1.0 + (b * (b * 4.0))
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.15e+24)
		tmp = a ^ 4.0;
	elseif (a <= 3.1e+14)
		tmp = Float64(-1.0 + Float64(b * Float64(b * 4.0)));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.15e+24)
		tmp = a ^ 4.0;
	elseif (a <= 3.1e+14)
		tmp = -1.0 + (b * (b * 4.0));
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.15e+24], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 3.1e+14], N[(-1.0 + N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.15e24 or 3.1e14 < a

   1. Initial program 41.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ 41.2%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)} \]

      2. sqr-pow 41.2%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      3. sqr-pow 41.2%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      4. fma-def 41.2%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      5. distribute-lft-in 41.2%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right)} - 1\right) \]

      6. sqr-neg 41.2%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\color{blue}{\left(\left(-b\right) \cdot \left(-b\right)\right)} \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\right) \]

      7. distribute-lft-in 41.2%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(\left(-b\right) \cdot \left(-b\right)\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right) \]

   3. Simplified 44.1%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right)} \]

   4. Taylor expanded in a around inf 94.5%

      \[\leadsto \color{blue}{{a}^{4}} \]

   if -1.15e24 < a < 3.1e14

   1. Initial program 97.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ 97.9%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)} \]

      2. sqr-pow 97.9%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      3. sqr-pow 97.9%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      4. fma-def 97.9%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      5. distribute-lft-in 97.9%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right)} - 1\right) \]

      6. sqr-neg 97.9%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\color{blue}{\left(\left(-b\right) \cdot \left(-b\right)\right)} \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\right) \]

      7. distribute-lft-in 97.9%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(\left(-b\right) \cdot \left(-b\right)\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right) \]

   3. Simplified 97.9%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right)} \]

   4. Taylor expanded in a around 0 98.0%

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   5. Step-by-step derivation

      1. associate--l+ 98.0%

         \[\leadsto \color{blue}{4 \cdot {b}^{2} + \left({b}^{4} - 1\right)} \]

      2. fma-def 98.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(4, {b}^{2}, {b}^{4} - 1\right)} \]

      3. unpow2 98.0%

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{b \cdot b}, {b}^{4} - 1\right) \]

      4. sub-neg 98.0%

         \[\leadsto \mathsf{fma}\left(4, b \cdot b, \color{blue}{{b}^{4} + \left(-1\right)}\right) \]

      5. metadata-eval 98.0%

         \[\leadsto \mathsf{fma}\left(4, b \cdot b, {b}^{4} + \color{blue}{-1}\right) \]

   6. Simplified 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, {b}^{4} + -1\right)} \]

   7. Taylor expanded in b around 0 74.9%

      \[\leadsto \color{blue}{4 \cdot {b}^{2} - 1} \]
   8. Step-by-step derivation

      1. fma-neg 74.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(4, {b}^{2}, -1\right)} \]

      2. unpow2 74.9%

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{b \cdot b}, -1\right) \]

      3. metadata-eval 74.9%

         \[\leadsto \mathsf{fma}\left(4, b \cdot b, \color{blue}{-1}\right) \]

   9. Simplified 74.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, -1\right)} \]
   10. Step-by-step derivation

       1. fma-udef 74.9%

          \[\leadsto \color{blue}{4 \cdot \left(b \cdot b\right) + -1} \]

       2. *-commutative 74.9%

          \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot 4} + -1 \]

       3. associate-*l* 74.9%

          \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right)} + -1 \]

   11. Applied egg-rr 74.9%

       \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right) + -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{+24}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{+14}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 6: 53.4% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.33:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot -12\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.33) (+ (* a (* (* b b) -12.0)) -1.0) (+ (* b (* b 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.33) {
		tmp = (a * ((b * b) * -12.0)) + -1.0;
	} else {
		tmp = (b * (b * 4.0)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.33d0)) then
        tmp = (a * ((b * b) * (-12.0d0))) + (-1.0d0)
    else
        tmp = (b * (b * 4.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.33) {
		tmp = (a * ((b * b) * -12.0)) + -1.0;
	} else {
		tmp = (b * (b * 4.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.33:
		tmp = (a * ((b * b) * -12.0)) + -1.0
	else:
		tmp = (b * (b * 4.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.33)
		tmp = Float64(Float64(a * Float64(Float64(b * b) * -12.0)) + -1.0);
	else
		tmp = Float64(Float64(b * Float64(b * 4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.33)
		tmp = (a * ((b * b) * -12.0)) + -1.0;
	else
		tmp = (b * (b * 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.33], N[(N[(a * N[(N[(b * b), $MachinePrecision] * -12.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.330000000000000016

   1. Initial program 30.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around 0 35.9%

      \[\leadsto \color{blue}{\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right)} - 1 \]
   3. Step-by-step derivation

      1. +-commutative 35.9%

         \[\leadsto \color{blue}{\left(\left(4 \cdot {b}^{2} + {b}^{4}\right) + -12 \cdot \left(a \cdot {b}^{2}\right)\right)} - 1 \]

      2. +-commutative 35.9%

         \[\leadsto \left(\color{blue}{\left({b}^{4} + 4 \cdot {b}^{2}\right)} + -12 \cdot \left(a \cdot {b}^{2}\right)\right) - 1 \]

      3. associate-+l+ 35.9%

         \[\leadsto \color{blue}{\left({b}^{4} + \left(4 \cdot {b}^{2} + -12 \cdot \left(a \cdot {b}^{2}\right)\right)\right)} - 1 \]

      4. associate-*r* 35.9%

         \[\leadsto \left({b}^{4} + \left(4 \cdot {b}^{2} + \color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}}\right)\right) - 1 \]

      5. *-commutative 35.9%

         \[\leadsto \left({b}^{4} + \left(4 \cdot {b}^{2} + \color{blue}{\left(a \cdot -12\right)} \cdot {b}^{2}\right)\right) - 1 \]

      6. metadata-eval 35.9%

         \[\leadsto \left({b}^{4} + \left(4 \cdot {b}^{2} + \left(a \cdot \color{blue}{\left(-3 \cdot 4\right)}\right) \cdot {b}^{2}\right)\right) - 1 \]

      7. associate-*l* 35.9%

         \[\leadsto \left({b}^{4} + \left(4 \cdot {b}^{2} + \color{blue}{\left(\left(a \cdot -3\right) \cdot 4\right)} \cdot {b}^{2}\right)\right) - 1 \]

      8. *-commutative 35.9%

         \[\leadsto \left({b}^{4} + \left(4 \cdot {b}^{2} + \left(\color{blue}{\left(-3 \cdot a\right)} \cdot 4\right) \cdot {b}^{2}\right)\right) - 1 \]

      9. distribute-rgt-in 35.9%

         \[\leadsto \left({b}^{4} + \color{blue}{{b}^{2} \cdot \left(4 + \left(-3 \cdot a\right) \cdot 4\right)}\right) - 1 \]

      10. unpow2 35.9%

          \[\leadsto \left({b}^{4} + \color{blue}{\left(b \cdot b\right)} \cdot \left(4 + \left(-3 \cdot a\right) \cdot 4\right)\right) - 1 \]

      11. metadata-eval 35.9%

          \[\leadsto \left({b}^{4} + \left(b \cdot b\right) \cdot \left(\color{blue}{1 \cdot 4} + \left(-3 \cdot a\right) \cdot 4\right)\right) - 1 \]

      12. distribute-rgt-in 35.9%

          \[\leadsto \left({b}^{4} + \left(b \cdot b\right) \cdot \color{blue}{\left(4 \cdot \left(1 + -3 \cdot a\right)\right)}\right) - 1 \]

      13. associate-*l* 35.9%

          \[\leadsto \left({b}^{4} + \color{blue}{b \cdot \left(b \cdot \left(4 \cdot \left(1 + -3 \cdot a\right)\right)\right)}\right) - 1 \]

      14. distribute-lft-in 35.9%

          \[\leadsto \left({b}^{4} + b \cdot \left(b \cdot \color{blue}{\left(4 \cdot 1 + 4 \cdot \left(-3 \cdot a\right)\right)}\right)\right) - 1 \]

      15. metadata-eval 35.9%

          \[\leadsto \left({b}^{4} + b \cdot \left(b \cdot \left(\color{blue}{4} + 4 \cdot \left(-3 \cdot a\right)\right)\right)\right) - 1 \]

      16. associate-*r* 35.9%

          \[\leadsto \left({b}^{4} + b \cdot \left(b \cdot \left(4 + \color{blue}{\left(4 \cdot -3\right) \cdot a}\right)\right)\right) - 1 \]

      17. metadata-eval 35.9%

          \[\leadsto \left({b}^{4} + b \cdot \left(b \cdot \left(4 + \color{blue}{-12} \cdot a\right)\right)\right) - 1 \]

   4. Simplified 35.9%

      \[\leadsto \color{blue}{\left({b}^{4} + b \cdot \left(b \cdot \left(4 + -12 \cdot a\right)\right)\right)} - 1 \]
   5. Step-by-step derivation

      1. +-commutative 35.9%

         \[\leadsto \color{blue}{\left(b \cdot \left(b \cdot \left(4 + -12 \cdot a\right)\right) + {b}^{4}\right)} - 1 \]

      2. associate-*r* 35.9%

         \[\leadsto \left(\color{blue}{\left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right)} + {b}^{4}\right) - 1 \]

      3. sqr-pow 35.9%

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right) + \color{blue}{{b}^{\left(\frac{4}{2}\right)} \cdot {b}^{\left(\frac{4}{2}\right)}}\right) - 1 \]

      4. metadata-eval 35.9%

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right) + {b}^{\color{blue}{2}} \cdot {b}^{\left(\frac{4}{2}\right)}\right) - 1 \]

      5. pow2 35.9%

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right) + \color{blue}{\left(b \cdot b\right)} \cdot {b}^{\left(\frac{4}{2}\right)}\right) - 1 \]

      6. metadata-eval 35.9%

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right) + \left(b \cdot b\right) \cdot {b}^{\color{blue}{2}}\right) - 1 \]

      7. pow2 35.9%

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(4 + -12 \cdot a\right) + \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) - 1 \]

      8. distribute-lft-out 35.9%

         \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(\left(4 + -12 \cdot a\right) + b \cdot b\right)} - 1 \]

      9. +-commutative 35.9%

         \[\leadsto \left(b \cdot b\right) \cdot \left(\color{blue}{\left(-12 \cdot a + 4\right)} + b \cdot b\right) - 1 \]

      10. fma-def 35.9%

          \[\leadsto \left(b \cdot b\right) \cdot \left(\color{blue}{\mathsf{fma}\left(-12, a, 4\right)} + b \cdot b\right) - 1 \]

   6. Applied egg-rr 35.9%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(-12, a, 4\right) + b \cdot b\right)} - 1 \]

   7. Taylor expanded in a around inf 34.3%

      \[\leadsto \color{blue}{-12 \cdot \left(a \cdot {b}^{2}\right)} - 1 \]
   8. Step-by-step derivation

      1. unpow2 34.3%

         \[\leadsto -12 \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right) - 1 \]

      2. associate-*r* 34.3%

         \[\leadsto \color{blue}{\left(-12 \cdot a\right) \cdot \left(b \cdot b\right)} - 1 \]

      3. *-commutative 34.3%

         \[\leadsto \color{blue}{\left(a \cdot -12\right)} \cdot \left(b \cdot b\right) - 1 \]

      4. associate-*l* 34.3%

         \[\leadsto \color{blue}{a \cdot \left(-12 \cdot \left(b \cdot b\right)\right)} - 1 \]

   9. Simplified 34.3%

      \[\leadsto \color{blue}{a \cdot \left(-12 \cdot \left(b \cdot b\right)\right)} - 1 \]

   if -0.330000000000000016 < a

   1. Initial program 87.4%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ 87.4%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)} \]

      2. sqr-pow 87.4%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      3. sqr-pow 87.4%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      4. fma-def 87.4%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      5. distribute-lft-in 87.4%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right)} - 1\right) \]

      6. sqr-neg 87.4%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\color{blue}{\left(\left(-b\right) \cdot \left(-b\right)\right)} \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\right) \]

      7. distribute-lft-in 87.4%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(\left(-b\right) \cdot \left(-b\right)\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right) \]

   3. Simplified 87.4%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right)} \]

   4. Taylor expanded in a around 0 82.8%

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   5. Step-by-step derivation

      1. associate--l+ 82.8%

         \[\leadsto \color{blue}{4 \cdot {b}^{2} + \left({b}^{4} - 1\right)} \]

      2. fma-def 82.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(4, {b}^{2}, {b}^{4} - 1\right)} \]

      3. unpow2 82.8%

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{b \cdot b}, {b}^{4} - 1\right) \]

      4. sub-neg 82.8%

         \[\leadsto \mathsf{fma}\left(4, b \cdot b, \color{blue}{{b}^{4} + \left(-1\right)}\right) \]

      5. metadata-eval 82.8%

         \[\leadsto \mathsf{fma}\left(4, b \cdot b, {b}^{4} + \color{blue}{-1}\right) \]

   6. Simplified 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, {b}^{4} + -1\right)} \]

   7. Taylor expanded in b around 0 63.3%

      \[\leadsto \color{blue}{4 \cdot {b}^{2} - 1} \]
   8. Step-by-step derivation

      1. fma-neg 63.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(4, {b}^{2}, -1\right)} \]

      2. unpow2 63.3%

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{b \cdot b}, -1\right) \]

      3. metadata-eval 63.3%

         \[\leadsto \mathsf{fma}\left(4, b \cdot b, \color{blue}{-1}\right) \]

   9. Simplified 63.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, -1\right)} \]
   10. Step-by-step derivation

       1. fma-udef 63.3%

          \[\leadsto \color{blue}{4 \cdot \left(b \cdot b\right) + -1} \]

       2. *-commutative 63.3%

          \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot 4} + -1 \]

       3. associate-*l* 63.3%

          \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right)} + -1 \]

   11. Applied egg-rr 63.3%

       \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right) + -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.33:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot -12\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \end{array} \]

Alternative 7: 37.3% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.00155:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 0.00155) -1.0 (* b (* b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.00155) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 0.00155d0) then
        tmp = -1.0d0
    else
        tmp = b * (b * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.00155) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.00155:
		tmp = -1.0
	else:
		tmp = b * (b * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.00155)
		tmp = -1.0;
	else
		tmp = Float64(b * Float64(b * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.00155)
		tmp = -1.0;
	else
		tmp = b * (b * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.00155], -1.0, N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.00154999999999999995

   1. Initial program 75.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ 75.2%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)} \]

      2. sqr-pow 75.2%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      3. sqr-pow 75.2%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      4. fma-def 75.2%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      5. distribute-lft-in 75.2%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right)} - 1\right) \]

      6. sqr-neg 75.2%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\color{blue}{\left(\left(-b\right) \cdot \left(-b\right)\right)} \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\right) \]

      7. distribute-lft-in 75.2%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(\left(-b\right) \cdot \left(-b\right)\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right) \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right)} \]

   4. Taylor expanded in a around 0 66.7%

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   5. Step-by-step derivation

      1. associate--l+ 66.7%

         \[\leadsto \color{blue}{4 \cdot {b}^{2} + \left({b}^{4} - 1\right)} \]

      2. fma-def 66.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(4, {b}^{2}, {b}^{4} - 1\right)} \]

      3. unpow2 66.7%

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{b \cdot b}, {b}^{4} - 1\right) \]

      4. sub-neg 66.7%

         \[\leadsto \mathsf{fma}\left(4, b \cdot b, \color{blue}{{b}^{4} + \left(-1\right)}\right) \]

      5. metadata-eval 66.7%

         \[\leadsto \mathsf{fma}\left(4, b \cdot b, {b}^{4} + \color{blue}{-1}\right) \]

   6. Simplified 66.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, {b}^{4} + -1\right)} \]

   7. Taylor expanded in b around 0 35.4%

      \[\leadsto \color{blue}{-1} \]

   if 0.00154999999999999995 < b

   1. Initial program 74.1%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ 74.1%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)} \]

      2. sqr-pow 74.1%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      3. sqr-pow 74.1%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      4. fma-def 74.1%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

      5. distribute-lft-in 74.1%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right)} - 1\right) \]

      6. sqr-neg 74.1%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\color{blue}{\left(\left(-b\right) \cdot \left(-b\right)\right)} \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\right) \]

      7. distribute-lft-in 74.1%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(\left(-b\right) \cdot \left(-b\right)\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right) \]

   3. Simplified 77.2%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right)} \]

   4. Taylor expanded in a around 0 81.3%

      \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
   5. Step-by-step derivation

      1. associate--l+ 81.3%

         \[\leadsto \color{blue}{4 \cdot {b}^{2} + \left({b}^{4} - 1\right)} \]

      2. fma-def 81.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(4, {b}^{2}, {b}^{4} - 1\right)} \]

      3. unpow2 81.3%

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{b \cdot b}, {b}^{4} - 1\right) \]

      4. sub-neg 81.3%

         \[\leadsto \mathsf{fma}\left(4, b \cdot b, \color{blue}{{b}^{4} + \left(-1\right)}\right) \]

      5. metadata-eval 81.3%

         \[\leadsto \mathsf{fma}\left(4, b \cdot b, {b}^{4} + \color{blue}{-1}\right) \]

   6. Simplified 81.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, {b}^{4} + -1\right)} \]

   7. Taylor expanded in b around 0 49.3%

      \[\leadsto \color{blue}{4 \cdot {b}^{2} - 1} \]
   8. Step-by-step derivation

      1. fma-neg 49.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(4, {b}^{2}, -1\right)} \]

      2. unpow2 49.3%

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{b \cdot b}, -1\right) \]

      3. metadata-eval 49.3%

         \[\leadsto \mathsf{fma}\left(4, b \cdot b, \color{blue}{-1}\right) \]

   9. Simplified 49.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, -1\right)} \]

   10. Taylor expanded in b around inf 49.2%

       \[\leadsto \color{blue}{4 \cdot {b}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 49.2%

          \[\leadsto 4 \cdot \color{blue}{\left(b \cdot b\right)} \]

       2. *-commutative 49.2%

          \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot 4} \]

       3. associate-*r* 49.2%

          \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right)} \]

   12. Simplified 49.2%

       \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.00155:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 8: 50.1% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ -1 + b \cdot \left(b \cdot 4\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (+ -1.0 (* b (* b 4.0))))

C

double code(double a, double b) {
	return -1.0 + (b * (b * 4.0));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (-1.0d0) + (b * (b * 4.0d0))
end function

Java

public static double code(double a, double b) {
	return -1.0 + (b * (b * 4.0));
}

Python

def code(a, b):
	return -1.0 + (b * (b * 4.0))

Julia

function code(a, b)
	return Float64(-1.0 + Float64(b * Float64(b * 4.0)))
end

MATLAB

function tmp = code(a, b)
	tmp = -1.0 + (b * (b * 4.0));
end

Wolfram

code[a_, b_] := N[(-1.0 + N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
2. Step-by-step derivation

   1. associate--l+ 74.9%

      \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)} \]

   2. sqr-pow 74.9%

      \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

   3. sqr-pow 74.9%

      \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

   4. fma-def 74.9%

      \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

   5. distribute-lft-in 74.9%

      \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right)} - 1\right) \]

   6. sqr-neg 74.9%

      \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\color{blue}{\left(\left(-b\right) \cdot \left(-b\right)\right)} \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\right) \]

   7. distribute-lft-in 74.9%

      \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(\left(-b\right) \cdot \left(-b\right)\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right) \]

3. Simplified 76.1%

   \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right)} \]

4. Taylor expanded in a around 0 70.5%

   \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
5. Step-by-step derivation

   1. associate--l+ 70.5%

      \[\leadsto \color{blue}{4 \cdot {b}^{2} + \left({b}^{4} - 1\right)} \]

   2. fma-def 70.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {b}^{2}, {b}^{4} - 1\right)} \]

   3. unpow2 70.5%

      \[\leadsto \mathsf{fma}\left(4, \color{blue}{b \cdot b}, {b}^{4} - 1\right) \]

   4. sub-neg 70.5%

      \[\leadsto \mathsf{fma}\left(4, b \cdot b, \color{blue}{{b}^{4} + \left(-1\right)}\right) \]

   5. metadata-eval 70.5%

      \[\leadsto \mathsf{fma}\left(4, b \cdot b, {b}^{4} + \color{blue}{-1}\right) \]

6. Simplified 70.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, {b}^{4} + -1\right)} \]

7. Taylor expanded in b around 0 54.1%

   \[\leadsto \color{blue}{4 \cdot {b}^{2} - 1} \]
8. Step-by-step derivation

   1. fma-neg 54.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {b}^{2}, -1\right)} \]

   2. unpow2 54.1%

      \[\leadsto \mathsf{fma}\left(4, \color{blue}{b \cdot b}, -1\right) \]

   3. metadata-eval 54.1%

      \[\leadsto \mathsf{fma}\left(4, b \cdot b, \color{blue}{-1}\right) \]

9. Simplified 54.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, -1\right)} \]
10. Step-by-step derivation

    1. fma-udef 54.1%

       \[\leadsto \color{blue}{4 \cdot \left(b \cdot b\right) + -1} \]

    2. *-commutative 54.1%

       \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot 4} + -1 \]

    3. associate-*l* 54.1%

       \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right)} + -1 \]

11. Applied egg-rr 54.1%

    \[\leadsto \color{blue}{b \cdot \left(b \cdot 4\right) + -1} \]

12. Final simplification 54.1%

    \[\leadsto -1 + b \cdot \left(b \cdot 4\right) \]

Alternative 9: 24.4% accurate, 130.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 -1.0)

C

double code(double a, double b) {
	return -1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -1.0d0
end function

Java

public static double code(double a, double b) {
	return -1.0;
}

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

function tmp = code(a, b)
	tmp = -1.0;
end

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 74.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
2. Step-by-step derivation

   1. associate--l+ 74.9%

      \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right)} \]

   2. sqr-pow 74.9%

      \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

   3. sqr-pow 74.9%

      \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

   4. fma-def 74.9%

      \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) - 1\right) \]

   5. distribute-lft-in 74.9%

      \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right)} - 1\right) \]

   6. sqr-neg 74.9%

      \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right)\right) + 4 \cdot \left(\color{blue}{\left(\left(-b\right) \cdot \left(-b\right)\right)} \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\right) \]

   7. distribute-lft-in 74.9%

      \[\leadsto {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(\color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(\left(-b\right) \cdot \left(-b\right)\right) \cdot \left(1 - 3 \cdot a\right)\right)} - 1\right) \]

3. Simplified 76.1%

   \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right)} \]

4. Taylor expanded in a around 0 70.5%

   \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right) - 1} \]
5. Step-by-step derivation

   1. associate--l+ 70.5%

      \[\leadsto \color{blue}{4 \cdot {b}^{2} + \left({b}^{4} - 1\right)} \]

   2. fma-def 70.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {b}^{2}, {b}^{4} - 1\right)} \]

   3. unpow2 70.5%

      \[\leadsto \mathsf{fma}\left(4, \color{blue}{b \cdot b}, {b}^{4} - 1\right) \]

   4. sub-neg 70.5%

      \[\leadsto \mathsf{fma}\left(4, b \cdot b, \color{blue}{{b}^{4} + \left(-1\right)}\right) \]

   5. metadata-eval 70.5%

      \[\leadsto \mathsf{fma}\left(4, b \cdot b, {b}^{4} + \color{blue}{-1}\right) \]

6. Simplified 70.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(4, b \cdot b, {b}^{4} + -1\right)} \]

7. Taylor expanded in b around 0 26.5%

   \[\leadsto \color{blue}{-1} \]

8. Final simplification 26.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023293 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))